A nonlocal finite element method for torsional statics and dynamics of circular nanostructures

• Presents a new nonlocal finite element model with numerical solutions.
• Solve the integral nonlocal stress relation instead of the differential approach.
• Show that the nonlocal effects do exist for all types of boundary conditions.

The torsional static and dynamic nonlocal effects for circular nanostructures subjected to concentrated and distributed torques are investigated based on the nonlocal elasticity stress theory. The total strain energy and kinetic energy components are derived and the variational energy principle is applied to derive the governing equation of motion and the corresponding boundary conditions. A new nonlocal finite element method (NL-FEM) is developed to solve the integral nonlocal equation. New numerical solutions for statics and dynamics of nonlocal nanoshafts, nanorods and nanotubes with various loads and boundary conditions are presented. The NL-FE numerical solutions are compared with analytical solutions obtained by solving the differential nonlocal equation. It is observed that the deformation angle as well as the ratio of nonlocal to classical deformation angle increases with increasing nonlocal nanoscale while the natural frequency for free torsional vibration decreases with increasing nanoscale. This paper concludes that the analytical nonlocal model and solutions, which apply the differential nonlocal constitutive relation, fails to capture the nonlocal boundary effects. The NL-FEM, which solves directly the original integral nonlocal stress relation, demonstrates nonlocal boundary effects for all cases of study. The differences of the differential and integral nonlocal stress relations are reported using representative numerical examples.